Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of evanescent waves to amplitude. This corresponds replacing interaction potential $v$ with an associated energy-dependent nonlocal ${\mathscr{V}}_k$ does not couple waves. The solutions $\psi(\mathbf{r})$ Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has remarkable property their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ momentum classical particle whose magnitude equals $k$. construct transfer matrix this class potentials and explore its representation terms evolution operator effective non-unitary quantum system. show above reduces first Born weak potentials, similarly semiclassical approximation, becomes valid at high energies. Furthermore, we identify infinite complex which is exact. also discuss appealing practical mathematical aspects scheme.